A concrete counterexample is presented as follows. Recall from [1]} that if \(e\in H\)  is an central idempotent
such that \(e\otimes e=(e\otimes 1)\Delta (e)=\Delta (e)(e\otimes 1)\)  and \(\varepsilon (e)=1\) .
A partial \(H\) -coaction on \(A = k\)  is given by \(\rho ^{l}(x)=e\otimes x\in H\otimes _k k\)  and \(\rho ^{r}(x)=x\otimes e\in k\otimes _k H\) .
It is easy to check that \(A\)  is a partial \(H\) -bicomodule algebra.
Therefore, \(A\)  is a partial \(H^{\ast }\) -bimodule algebra. So we can obtain the partial twisted smash product \(k \circledast H^{\ast }\) , here we only consider the element \(f\)  of \(H^{\ast }\)  and check the condition. Then we get
\(( a\leftharpoonup S^{\ast }(h_{(2)}))\otimes h_{(1)}= ( x\leftharpoonup S^{\ast }(f_{(2)}))\otimes f_{(1)}=x\otimes <f_{(2)}, e>f_{(1)}\)  and
\(&&( a\leftharpoonup S^{\ast }(h_{(1)}))\otimes h_{(2)}=( x\leftharpoonup S^{\ast }(f_{(1)}))\otimes f_{(2)} \\&&= <f_{(1)}, e>x\otimes f_{(2)}= x\otimes <f_{(1)}, e>f_{(2)}.\) 
